Question

find (limlimits_{x \to 7} \frac{x - 7}{x^2 - 49}).
select the correct choice below and, if necessary, fill in the answer box t
a. (limlimits_{x \to 7} \frac{x - 7}{x^2 - 49} = square)
(type an integer or a simplified fraction.)
b. the limit does not exist.

Explanation:

Step1: Factor the denominator

The denominator \(x^2 - 49\) is a difference of squares, so we can factor it as \((x - 7)(x + 7)\). So the expression becomes \(\lim_{x \to 7} \frac{x - 7}{(x - 7)(x + 7)}\).

Step2: Cancel the common factor

We can cancel out the common factor of \(x - 7\) (assuming \(x
eq 7\), which is valid for finding the limit as \(x\) approaches 7, not at \(x = 7\)). This gives us \(\lim_{x \to 7} \frac{1}{x + 7}\).

Step3: Substitute \(x = 7\)

Now we substitute \(x = 7\) into the simplified expression \(\frac{1}{x + 7}\). So we get \(\frac{1}{7 + 7} = \frac{1}{14}\).

Answer:

\(\frac{1}{14}\)